symmetric punching shear
3.6.1 Linear elastic materials
Wang [Wan96] defines Fracture Mechanics (FM) as a set of theories that describe the behavior of solids or structures with geometrical discontinuities at the scale of the struc- ture, as notches or cracks.
A notch can be defined as a geometric discontinuity which has a definite depth (D) and root radius (ρ), and a crack can be seen as a notch with a root radius ρ → 0. Let us consider the plate ofFigure 3.7containing a notch and uniaxially loaded (stress σ). The linear elastic analysis of this plate (analytically, numerically or experimentally) allows to study the severity of the notch, i.e., how disturbed the stress field is in the vicinity of the notch compared with the uniform distribution. The maximum applied stress σmax
is related with the nominal stress σN by the stress concentration factor KT, which can
be approximated for elliptical notch shapes as follows:
KT = σmax
σN = 1 + 2 s
D
ρ (3.23)
When ρ → 0, KT → ∞, and as a result σmax → ∞. The stress concentration factor is
thus not suitable to distinguish between different crack lengths and applied stress levels.
Figure 3.7: Notch on uniaxially loaded plate [Wan96]
The basic concept proposed by Griffith [Gri21] in 1921 to formulate the linear elastic theory of crack propagation is the following: whether a stressed cracked body remains stable or becomes unstable is dependent on whether the cracked body contains sufficient
energy to afford creating additional surface maintaining equilibrium. Let us consider the stressed cracked thin plate with a crack length of 2a and thickness B of Figure 3.8.
Figure 3.8: Stressed cracked plate [Wan96]
According to the law of conservation of energy the work performed per unit time by the applied loads ˙W must be equal to the rates of change of the internal elastic energy ˙UE,
plastic energy ˙UP and kinetic energy ˙K of the body, and the energy per unit time ˙Γ
spent in increasing the crack area: ˙
W = ˙UE+ ˙UP + ˙K + ˙Γ (3.24)
Assuming a slow crack growth, the kinetic energy can be neglected (K = ˙K = 0). Since all changes with respect to time are caused by changes in crack size, the differential operator with respect to time δ/δt can be given by:
δ δt = δ δA δA δt = ˙A δ δA (3.25)
where A = 2aB represents the crack area. However the total crack surface area is twice the crack area. ThereforeEquation 3.24 can be rewritten as:
−δΠ δA = δUP δA + δΓ δA (3.26) where Π = UE− W (3.27)
represents the potential energy of the system. Equation 3.26 shows that the reduction of the potential energy is equal to the energy dissipated in plastic work and surface creation.
For an ideally brittle material UP = 0, i.e., the dissipated energy in plastic energy
is negligible. The law of conservation of energy of Equation 3.26 can once again be rewritten as
−δΠ
δA =
δΓ
δA = 2γ (3.28)
Linear elastic fracture mechanics property) and factor 2 to the two new material surfaces formed during crack growth.
Equation 3.28obliges that sufficient available potential energy is available in the system to overcome the surface energy of the material in order to have crack growth. A crack- extension force G can be defined as
G= −δΠ
δA (3.29)
The total energy of the system can be defined as:
Utotal= (−W + UE) + Γ (3.30)
According to Clapeyron’s theorem of linear elastostatics that states that the potential energy of deformation of a body, which is in equilibrium under a given load, is equal to half the work done by the external forces, and assuming these forces remained constant from the initial state to the final state, we have:
W = 2UE (3.31)
The crack-extension force can be rewritten as:
G= δUE
δA (3.32)
and the total energy of the system:
Utotal = −UE+ Γ (3.33)
Griffith[Gri21] applied the stress solutions of Inglis[Ing13]to show that the increase in strain energy due to the elliptic cavity (zero radius) in an infinite plane of thickness B is given by:
UE =
πa2σ2B
E (3.34)
where E is Young’s modulus of the material. The energy spent increasing the crack area is given by:
Γ = 4aBγ (3.35)
Thus, the total system energy of the thin plate becomes
Utotal= −
πa2σ2B
E + 4aBγ (3.36)
which exhibits a maximum at the following crack length:
ac= 2γE
πσ2 (3.37)
creases with the applied stress. Alternatively, the critical stress level that the plate can sustain for a given crack length a is
σc= s
2Eγ
πa (3.38)
All the presented equations are only valid for linear elastic materials.
The energy release rate G characterises the amount of energy released if the crack ad- vances a unit length. When this value is greater than the surface energy of the material the crack can grow, otherwise no crack propagation is possible.
The energy release rate G can be determined from experimental tests. Figure 3.9(a)
presents a cracked specimen made of a linear elastic material subjected to a load P and/or a displacement u. When the crack length is a, the specimen is less compliant than when the crack length is a + δa. The compliance C of the specimen (geometry dependent) is given by:
C= u
P (3.39)
Figure 3.9: Edge stressed cracked plate and load displacement characteristics
[Wan96]: (a) geometry; (b) constant load crack extension; and (c) crack extension under constant displacement
If we consider a test performed at constant load conditions (Figure 3.9(b)), an increase of crack length of δa results in a potential energy change of δΠ (difference between the external work and the stored but recoverable elastic strain energy). The increase of elastic strain energy δUE is given by:
δUE = 12P1u2−12P1u1 (3.40)
and the work performed by the load P in the distance (u2− u1) as:
δW = P1(u2− u1) (3.41)
The energy spent in increasing crack surfaces is given by: − δΠ = δW − δUE = P1(u2− u1) − 1 2P1(u2− u1) = 1 2P1(u2− u1) = 1 2P1δu (3.42)
Linear elastic fracture mechanics which means that the energy spent in crack development was supplied by the work of the external load.
In a similar way, if we consider a test performed at constant displacement conditions (Figure 3.9(c)), an increase in crack length causes a decrease in the stored elastic strain energy of:
δUE = 12(P1− P2)u1= 12u1δP (3.43)
which is spent in increasing crack surface since no external work is done.
When an increase in crack area δA tends to zero, the compliance C is the same for both constant load and constant displacement conditions, which means that the difference between the energy spent in crack growth in both cases tends to zero:
1 2P1δu= 1 2u1δP ⇔ δu= u PδP = CδP (3.44)
and the energy release for both cases is given by:
G= 1
2CP δP (3.45)
The strain or potential energy release rate with respect to crack length for small crack area increases δA = Bδa can be found experimentally in a plate of uniform thickness B as: G= 1 2P δu δA = P2 2 δC δA (3.46)
G can be determined by means of measurements of the compliance of a specimen with
different crack lengths.
There are three different stressing modes of a crack. The opening mode or mode I (Figure 3.10(a)) corresponds to normal separation of crack surfaces under tensile stresses. The sliding mode or mode II corresponds to a crack propagation normal to the crack front under shear stresses. Finally, the tearing mode or mode III corresponds to a crack propagation parallel to the crack front, also when subjected to shear stresses. A stressed body can experience one or a combination of these modes.
Figure 3.10: Stressing modes of a crack [Wan96]: (a) opening mode or mode I; (b) sliding mode or mode II; and (c) tearing mode or mode III
The stress, strain and displacement fields of a cracked linear elastic body can be calcu- lated analytically by using the Westergaard method [Wes39], for example. Solving this problem yields these fields in the vicinity of a crack as a linear proportional function of the stress intensity factors K, which embody the loading and geometry conditions. The stress intensity factors and the searched fields can be found in many handbooks. In general the stress intensity factor depends on the applied stress, crack size and geometry:
K = Y σ√πa (3.47)
where Y is the so-called geometry factor, dependent of the body and crack geometries. For a center crack in an infinite plate, Y = 1. Y can be found in many handbooks for various practical situations, but in general, Y can be obtained from linear elastic finite element analysis.
As the stress and displacement fields are linearly proportional to the stress intensity factor, the superposition principle is applicable in LEFM.
Let us now recall Griffith’s energy concept to show that it is related with the stress intensity factor. Figure 3.11shows the forces needed to close a crack over an infinitesimal distance δ in a plate of thickness B. The work done by these forces is equal to the energy needed to make the crack grow this δ distance:
δUE = 2B Z 0 δ1 2σyuydx = 2B Z a a−δ 1 2σyuydx (3.48)
where factor 2 is related to the two surfaces of the crack and factor 1/2 is the assumed proportionality between tractions and the corresponding displacement uy. Using the
expressions for σy and uy that can be found in many handbooks and adopting polar
coordinates with origin at the crack tip (r,θ), we get:
G= lim δ→0 2K2 I πEδ Z 0 δsδ − r r dr = limδ→0 2K2 I πE Z 0 π/2scos2θ
sin2θ2δ sin θ cos θdθ (3.49)
yielding:
G= K
2
I
E (3.50)
for plane stress conditions and:
G= K
2
I
E (1 − ν
2) (3.51)
for plane strain conditions. Subscript I stands for mode I.